DOCSLIB.ORG

Subgraph Complementation Fedor V

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Subgraph Complementation Fedor V Subgraph Complementation Fedor V. Fomin, Petr A. Golovach, Torstein Strømme, Dimitrios M. Thilikos To cite this version: Fedor V. Fomin, Petr A. Golovach, Torstein Strømme, Dimitrios M. Thilikos. Subgraph Comple- mentation. Algorithmica, Springer Verlag, 2020, 82 (7), pp.1859-1880. 10.1007/s00453-020-00677-8. hal-03002656 HAL Id: hal-03002656 https://hal.archives-ouvertes.fr/hal-03002656 Submitted on 20 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Algorithmica https://doi.org/10.1007/s00453-020-00677-8 Subgraph Complementation Fedor V. Fomin1 · Petr A. Golovach1 · Torstein J. F. Strømme1 Dimitrios M. Thilikos2 Received: 30 January 2019 / Accepted: 8 January 2020 © The Author(s) 2020 Abstract A subgraph complement of the graph G is a graph obtained from G by complement- ing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G , is there a subgraph complement of G which is in G ? We show that this problem can be solved in polynomial time for various choices of the graphs class G , such as bipartite, d-degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of regular graphs. Keywords Subgraph complementation · Graph editing · Graph classes Mathematics Subject Classifcation 05C85 · 68Q25 1 Introduction One of the most important questions in graph theory concerns the efciency of rec- ognition of a graph class G . For example, how fast we can decide whether a graph is chordal, 2-connected, triangle-free, of bounded treewidth, bipartite, 3-color- able, or excludes some fxed graph as a minor? In particular, the recent develop- ments in parameterized algorithms are partially driven by the problems of recog- nizing of graph classes which difer only up to a “small disturbance” from graph * Torstein J. F. Strømme [email protected] Fedor V. Fomin [email protected] Petr A. Golovach [email protected] Dimitrios M. Thilikos [email protected] 1 University of Bergen, Bergen, Norway 2 LIRMM, Univ. Montpellier, CNRS, Montpellier, France Algorithmica classes recognizable in polynomial time. The amount of disturbance is quantifed in “atomic” operations required for modifying an input graph into the “well-behaving” graph class G . The standard operations could be edge/vertex deletions, additions or edge contractions. Many problems in graph algorithms fall into this graph modifca- tion category: is it possible to add at most k edges to make a graph 2-edge connected or to make it chordal? Or is it possible to delete at most k vertices such that the resulting graph has no edges or contains no cycles? A rich subclass of modifcation problems concerns edge editing problems. Here the “atomic” operation is the change of adjacency, i.e. for a pair of vertices u, v, we can either add an edge uv or delete the edge uv. For example, the CLUSTER EDITING problem asks to transform an input graph into a cluster graph—that is, a disjoint union of cliques—by fipping at most k adjacency relations. Besides the basic edge editing, it is natural to consider problems where the set of removed and added edges should satisfy some structural constraints. In particular, such problems were considered for complementation problems. Recall that the com- plement of a graph G is a graph H on the same vertices such that two distinct verti- ces of H are adjacent if and only if they are not adjacent in G. Seidel (see [22–24]) introduced the operation that is now known as the Seidel switch. For a vertex v of a graph G, this operation complements the adjacencies of v, that is, it removes the edges incident to v and makes v adjacent to the non-neighbors of v in G. Seidel switching a set of vertices U entails consecutively switching each vertex in the set. The result is that all adjacencies between U and its complement V(G)⧵U are fipped. The study of the algorithmic question whether it is possible to obtain a graph from a given graph class by the Seidel switch was initiated by Ehrenfeucht et al. [7]. Fur- ther results were established in [14–16, 18, 19]. Another important operation of this type is the local complementation. For a ver- tex v of a graph G, the local complementation of G at v is the graph obtained from G by replacing G[N(v)] by its complement. This operation plays crucial role in the def- inition of vertex-minors [20] and was investigated in this context (see, e.g. [6, 21]). See also [2, 17] for some algorithmic results concerning local complementations. In this paper we study the subgraph complement of a graph, which was intro- duced by Kamiński et al. [17] in their study of the clique-width of a graph. A sub- graph complement of a graph G is a graph obtained from G by complementing all the edges of one of its induced subgraphs. More formally, for a graph G and S ⊆ V(G) , we defne G ⊕ S as the graph with the vertex set V(G) whose edge set is defned as follows: a pair of distinct vertices u, v is an edge of G ⊕ S if and only if one of the following holds: • uv ∈ E(G)∧(u ∉ S ∨ v ∉ S) , or • uv ∉ E(G)∧u ∈ S ∧ v ∈ S. Thus when the set S consists only of two vertices {u, v} , the operation simply changes the adjacency between u and v, and for a larger set S, G ⊕ S changes the adjacency relations for all pairs of vertices of S. We say that a graph H is a subgraph complement of the graph G if H is isomor- phic to G ⊕ S for some S ⊆ V(G) . For a graph class G and a graph G, we say that Algorithmica there is a subgraph complement of G to G if for some S ⊆ V(G) , we have G ⊕ S ∈ G . (1) We denote by G the class of graphs such that its members can be subgraph com- plemented to G. Let G be a graph class. We consider the following generic algorithmic problem. (1) In other words, how difcult is it to recognize the class G ? In this paper we (1) show that there are many well-known graph classes G such that G is recognizable in polynomial time. We show that G O ⋅ 4 6 G • SUBGRAPH COMPLEMENT TO is solvable in (f (n) n + n ) time when is a triangle-free graph class recognizable in f(n) time. For example, this implies that (1) when G is the class of bipartite graphs, the class G is recognizable in polyno- mial time. This result is found in Sect. 3. G ⋅ 2O(d) G • SUBGRAPH COMPLEMENT TO is solvable in f (n) n time when is a d-degen- erate graph class recognizable in f(n) time. Thus when G is the class of planar graphs, class of cubic graphs, class of graph of bounded treewidth, or class of (1) H-minor free graphs, then the class G is recognizable in polynomial time. This result is found in Sect. 4. G G • SUBGRAPH COMPLEMENT TO is solvable in polynomial time when is a class of bounded clique-width expressible in monadic second-order logic (with no edge G set quantifcation). In particular, if is the class of P4-free graphs (cographs), (1) then G is recognizable in polynomial time. This result is found in Sect. 6. G G • SUBGRAPH COMPLEMENT TO is solvable in polynomial time when can be (1) described by a 2 × 2 M-partition matrix. Therefore G is recognizable in polyno- mial time when G is the class of split graphs, as they can be described by such a matrix. This result is found in Sect. 5. There are nevertheless cases when the problem is NP-hard. In particular, we prove that this holds when G is the class of regular graphs. This result is found in Sect. 7. 2 Preliminaries + We let ℕ∶={0, 1, 2 …} be the set of the natural numbers, and we let ℕ ∶=ℕ⧵{0} be the set of positive integers. For a set A and a non-negative integer k ∈ ℕ , we denote A A k ∶={A� ⊆ A ∣ A� = k} the family of all subsets of of size as k . Simple Graph A (simple) graph is a pair G∶=(V, E) , where V is a set of vertices V E ⊆ edges {u, v}∈E(G) uv and 2 is a set of . For an edge we use the shorthand Algorithmica (or equivalently, vu). For a given graph G, we refer to its vertex set as V(G), and its edge set as E(G). For a set of edges F ⊆ E(G) , we denote by V(F) the set of vertices that are contained in the edges of F, i.e. V(F)∶= uv∈F{u, v}. (Induced) Subgraph For graphs G and H we⋃ say that H is a subgraph of G, denoted by H ⊆ G if V(H) ⊆ V(G) and E(H) ⊆ E(G) .
Load more

Recommended publications
  • Symmetry and Structure of Graphs
    Symmetry and Structure of graphs by Kovács Máté Submitted to Central European University Department of Department of Mathematics and its Applications In partial fulfillment of the requirements for the degree of Master of Science Supervisor: Dr. Hegedus˝ Pál CEU eTD Collection Budapest, Hungary 2014 I, the undersigned [Kovács Máté], candidate for the degree of Master of Science at the Central European University Department of Mathematics and its Applications, declare herewith that the present thesis is exclusively my own work, based on my research and only such external information as properly credited in notes and bibliography. I declare that no unidentified and illegitimate use was made of work of others, and no part the thesis infringes on any person’s or institution’s copyright. I also declare that no part the thesis has been submitted in this form to any other institution of higher education for an academic degree. Budapest, 9 May 2014 ————————————————— Signature CEU eTD Collection c by Kovács Máté, 2014 All Rights Reserved. ii Abstract The thesis surveys results on structure and symmetry of graphs. Structure and symmetry of graphs can be handled by graph homomorphisms and graph automorphisms - the two approaches are compatible. Two graphs are called homomorphically equivalent if there is a graph homomorphism between the two graphs back and forth. Being homomorphically equivalent is an equivalence relation, and every class has a vertex minimal element called the graph core. It turns out that transitive graphs have transitive cores. The possibility of a structural result regarding transitive graphs is investigated. We speculate that almost all transitive graphs are cores.
    [Show full text]
  • Vertex Deletion Problems on Chordal Graphs∗†
    Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices.
    [Show full text]
  • A Comparison of Two Approaches for Polynomial Time Algorithms
    A comparison of two approaches for polynomial time algorithms computing basic graph parameters∗† Frank Gurski‡ March 26, 2018 Abstract In this paper we compare and illustrate the algorithmic use of graphs of bounded tree- width and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-width in polynomial time. We also present linear time algorithms for computing the latter four basic graph parameters on trees, i.e. graphs of tree-width 1, and on co-graphs, i.e. graphs of clique-width at most 2. Keywords:graph algorithms, graph parameters, clique-width, NLC-width, tree-width 1 Introduction A graph parameter is a mapping that associates every graph with a positive integer. Well known graph parameters are independence number, dominating number, and chromatic num- ber. In general the computation of such parameters for some given graph is NP-hard. In this work we give fixed-parameter tractable (fpt) algorithms for computing basic graph parameters restricted to graph classes of bounded tree-width and graph classes of bounded clique-width. The tree-width of graphs has been defined in 1976 by Halin [Hal76] and independently in arXiv:0806.4073v1 [cs.DS] 25 Jun 2008 1986 by Robertson and Seymour [RS86] by the existence of a tree decomposition. Intuitively, the tree-width of some graph G measures how far G differs from a tree. Two more powerful and more recent graph parameters are clique-width1 and NLC-width2 both defined in 1994, by Courcelle and Olariu [CO00] and by Wanke [Wan94], respectively.
    [Show full text]
  • Computing Directed Path-Width and Directed Tree-Width of Recursively
    Computing directed path-width and directed tree-width of recursively defined digraphs∗ Frank Gurski1 and Carolin Rehs1 1University of D¨usseldorf, Institute of Computer Science, Algorithmics for Hard Problems Group, 40225 D¨usseldorf, Germany June 13, 2018 Abstract In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digraphs. Since directed co-graphs are precisely those digraphs which can be defined by the disjoint union, order composition, and series composition our results imply the equality of directed path-width and directed tree-width for directed co-graphs and also a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs, which generalizes the known results for undirected co-graphs of Bodlaender and M¨ohring. Keywords: directed path-width; directed tree-width; directed co-graphs 1 Introduction Tree-width is a well-known graph parameter [36]. Many NP-hard graph problems admit poly- nomial-time solutions when restricted to graphs of bounded tree-width using the tree-decom- position [1, 3, 22, 27]. The same holds for path-width [35] since a path-decomposition can be regarded as a special case of a tree-decomposition.
    [Show full text]
  • Online Graph Coloring
    Online Graph Coloring Jinman Zhao - CSC2421 Online Graph coloring Input sequence: Output: Goal: Minimize k. k is the number of color used. Chromatic number: Smallest number of need for coloring. Denoted as . Lower bound Theorem: For every deterministic online algorithm there exists a logn-colorable graph for which the algorithm uses at least 2n/logn colors. The performance ratio of any deterministic online coloring algorithm is at least . Transparent online coloring game Adversary strategy : The collection of all subsets of {1,2,...,k} of size k/2. Avail(vt): Admissible colors consists of colors not used by its pre-neighbors. Hue(b)={Corlor(vi): Bin(vi) = b}: hue of a bin is the set of colors of vertices in the bin. H: hue collection is a set of all nonempty hues. #bin >= n/(k/2) #color<=k ratio>=2n/(k*k) Lower bound Theorem: For every randomized online algorithm there exists a k- colorable graph on which the algorithm uses at least n/k bins, where k=O(logn). The performance ratio of any randomized online coloring algorithm is at least . Adversary strategy for randomized algo Relaxing the constraint - blocked input Theorem: The performance ratio of any randomized algorithm, when the input is presented in blocks of size , is . Relaxing other constraints 1. Look-ahead and bufferring 2. Recoloring 3. Presorting vertices by degree 4. Disclosing the adversary’s previous coloring First Fit Use the smallest numbered color that does not violate the coloring requirement Induced subgraph A induced subgraph is a subset of the vertices of a graph G together with any edges whose endpoints are both in the subset.
    [Show full text]
  • Strong Triadic Closure in Cographs and Graphs of Low Maximum Degree
    Strong Triadic Closure in Cographs and Graphs of Low Maximum Degree Athanasios L. Konstantinidis1, Stavros D. Nikolopoulos2, and Charis Papadopoulos1 1 Department of Mathematics, University of Ioannina, Greece. [email protected], [email protected] 2 Department of Computer Science & Engineering, University of Ioannina, Greece. [email protected] Abstract. The MaxSTC problem is an assignment of the edges with strong or weak labels having the maximum number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent. The Cluster Deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques. Both problems are known to be NP-hard and an optimal solution for the Cluster Deletion problem provides a solu- tion for the MaxSTC problem, however not necessarily an optimal one. In this work we give the first systematic study that reveals graph families for which the optimal solutions for MaxSTC and Cluster Deletion coincide. We first show that MaxSTC coincides with Cluster Dele- tion on cographs and, thus, MaxSTC is solvable in quadratic time on cographs. As a side result, we give an interesting computational charac- terization of the maximum independent set on the cartesian product of two cographs. Furthermore we study how low degree bounds influence the complexity of the MaxSTC problem. We show that this problem is polynomial-time solvable on graphs of maximum degree three, whereas MaxSTC becomes NP-complete on graphs of maximum degree four. The latter implies that there is no subexponential-time algorithm for MaxSTC unless the Exponential-Time Hypothesis fails.
    [Show full text]
  • Hyperbolicity and Complement of Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 1882–1887 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Hyperbolicity and complement of graphs Sergio Bermudo a, José M. Rodríguez b, José M. Sigarreta c,∗, Eva Tourís b a Departamento de Economía, Métodos cuantitativos e Historia Económica, Universidad Pablo de Olavide, Carretera de Utrera Km. 1, 41013 Sevilla, Spain b Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain c Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650 Acalpulco Gro., Mexico article info a b s t r a c t Article history: If X is a geodesic metric space and x1; x2; x3 2 X, a geodesic triangle T D fx1; x2; x3g is the Received 20 July 2010 union of the three geodesics Tx1x2U, Tx2x3U and Tx3x1U in X. The space X is δ-hyperbolic (in Received in revised form 29 April 2011 the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two Accepted 11 May 2011 other sides, for every geodesic triangle T in X. We denote by δ.X/ the sharp hyperbolicity constant of X, i.e. δ.X/ VD inffδ ≥ 0 V X is δ-hyperbolicg. The study of hyperbolic graphs is Keywords: an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the Graph hyperbolicity of a graph related to it.
    [Show full text]
  • An Introduction to Algebraic Graph Theory
    An Introduction to Algebraic Graph Theory Cesar O. Aguilar Department of Mathematics State University of New York at Geneseo Last Update: March 25, 2021 Contents 1 Graphs 1 1.1 What is a graph? ......................... 1 1.1.1 Exercises .......................... 3 1.2 The rudiments of graph theory .................. 4 1.2.1 Exercises .......................... 10 1.3 Permutations ........................... 13 1.3.1 Exercises .......................... 19 1.4 Graph isomorphisms ....................... 21 1.4.1 Exercises .......................... 30 1.5 Special graphs and graph operations .............. 32 1.5.1 Exercises .......................... 37 1.6 Trees ................................ 41 1.6.1 Exercises .......................... 45 2 The Adjacency Matrix 47 2.1 The Adjacency Matrix ...................... 48 2.1.1 Exercises .......................... 53 2.2 The coefficients and roots of a polynomial ........... 55 2.2.1 Exercises .......................... 62 2.3 The characteristic polynomial and spectrum of a graph .... 63 2.3.1 Exercises .......................... 70 2.4 Cospectral graphs ......................... 73 2.4.1 Exercises .......................... 84 3 2.5 Bipartite Graphs ......................... 84 3 Graph Colorings 89 3.1 The basics ............................. 89 3.2 Bounds on the chromatic number ................ 91 3.3 The Chromatic Polynomial .................... 98 3.3.1 Exercises ..........................108 4 Laplacian Matrices 111 4.1 The Laplacian and Signless Laplacian Matrices .........111 4.1.1
    [Show full text]
  • Complexity of the Cluster Deletion Problem on Subclasses of Chordal Graphs∗
    Complexity of the cluster deletion problem on subclasses of chordal graphs∗ Flavia Bonomoy Guillermo Duranz Mario Valencia-Pabonx Abstract We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been studied extensively. We investigate the complexity of the decision CD problem for the family of chordal graphs, showing that it is NP-complete for weighted split graphs, weighted interval graphs and unweighted chordal graphs. We also prove that the problem is NP-complete for weighted cographs. Some polynomial-time solvable cases of the optimization problem are also identified, in particular CD for unweighted split graphs, unweighted proper interval graphs and weighted block graphs. Keywords:Block graphs, Cliques, Edge-deletion, Cluster deletion, Interval graphs, Split graphs, Submodular functions, Chordal graphs, Cographs, NP-completeness. 1 Introduction Clustering is an important task in the data analysis process. It can be viewed as a data modeling technique that provides an attractive mechanism for automatically finding the hidden structure of large data sets. The input to the problem is typically a set of elements and pairwise similarity values between elements. The goal is to partition these elements into subsets called clusters such that two meta-criteria are satisfied: homogeneity{elements in a given cluster are highly similar to each other; and separation{elements from different clusters have low similarity to each other.
    [Show full text]
  • Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width∗
    MULTISCALE MODEL. SIMUL. c xxxx Society for Industrial and Applied Mathematics Vol. xx, No. x, pp. x–x ALMOST OPTIMAL LOWER BOUNDS FOR PROBLEMS PARAMETERIZED BY CLIQUE-WIDTH∗ FEDOR V. FOMIN† , PETR A. GOLOVACH† , DANIEL LOKSHTANOV† , AND SAKET SAURABH‡ Abstract. We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dom- inating Set problems on graphs of bounded clique-width. We show that on an n-vertex graph of clique-width t both problems • cannot be solved in time f(t)no(t) for any function f of t unless Exponential Time Hy- pothesis (ETH) fails, and • can be solved in time nO(t). Key words. Exponential Time Hypothesis, clique-width, max-cut, edge dominating set AMS subject classifications. 05C85, 68R10, 68Q17, 68Q25, 68W40 1. Introduction. Tree-width is one of the most fundamental parameters in graph algorithms. Graphs of bounded tree-width enjoy good algorithmic properties similar to trees and this is why many problems which are hard on general graphs can be solved efficiently when the input is restricted to graphs of bounded tree-width. On the other hand, many hard problems also become tractable when restricted to graphs “similar to complete graphs”. Courcelle and Olariu [6] introduced the notion of clique-width which captures nice algorithmic properties of both extremes. Since 2000, the research on algorithmic and structural aspects of clique-width is an active direction in Graph Algorithms, Logic, and Complexity. Corneil et al. [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. Fellows et al. [12] settled a long standing open problem by showing that computing clique-width is NP-hard.
    [Show full text]
  • On the Structure of Compact Graphs
    Opuscula Math. 37, no. 6 (2017), 875–886 http://dx.doi.org/10.7494/OpMath.2017.37.6.875 Opuscula Mathematica ON THE STRUCTURE OF COMPACT GRAPHS Reza Nikandish and Farzad Shaveisi Communicated by Mirko Horňák Abstract. A simple graph G is called a compact graph if G contains no isolated vertices and for each pair x, y of non-adjacent vertices of G, there is a vertex z with N(x) N(y) N(z), ∪ ⊆ where N(v) is the neighborhood of v, for every vertex v of G. In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph G, then the descending chain condition holds for the set of neighbors of G. Keywords: compact graph, vertex degree, cycle, neighborhood. Mathematics Subject Classification: 05C07, 05C38, 68R10. 1. INTRODUCTION Throughout this paper, a graph G is an undirected simple graph with the vertex set V = V (G) and the edge set E = E(G). If vertices x and y are adjacent we write x y. By G, we mean the complement graph of G. Define the neighborhood N (x) of − G a vertex x to be the set of all vertices adjacent to x and let N(z) = N(z) z denote ∪ { } the closure of NG(z). When there is no risk of confusion, we denote NG(v) by N(v).
    [Show full text]
  • Graph-Based Data Clustering with Overlaps
    Proc. 15th COCOON, 2009 Graph-Based Data Clustering with Overlaps Michael R. Fellows1⋆, Jiong Guo2, Christian Komusiewicz2⋆⋆, Rolf Niedermeier2, and Johannes Uhlmann2⋆ ⋆ ⋆ 1 PC Research Unit, Office of DVC (Research), University of Newcastle, Callaghan, NSW 2308, Australia. [email protected] 2 Institut f¨ur Informatik, Friedrich-Schiller-Universit¨at Jena Ernst-Abbe-Platz 2, D-07743 Jena, Germany {jiong.guo,c.komus,rolf.niedermeier,johannes.uhlmann}@uni-jena.de Abstract. We introduce overlap cluster graph modification problems where, other than in most previous work, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a mini- mum number of edge modifications such that the resulting graph consists of clusters (maximal cliques) that may overlap up to a certain amount specified by the overlap number s. In the case of s-vertex overlap, each vertex may be part of at most s maximal cliques; s-edge overlap is anal- ogously defined in terms of edges. We provide a complete complexity dichotomy (polynomial-time solvable vs NP-complete) for the underly- ing edge modification problems, develop forbidden subgraph characteri- zations of “cluster graphs with overlaps”, and study the parameterized complexity in terms of the number of allowed edge modifications, achiev- ing fixed-parameter tractability results (in case of constant s-values) and parameterized hardness (in case of unbounded s-values). 1 Introduction Graph-based data clustering is an important tool in exploratory data analy- sis [21,25]. The applications range from bioinformatics [2,22] to image process- ing [24]. The formulation as a graph-theoretic problem relies on the notion of a similarity graph, where vertices represent data items and an edge between two vertices expresses high similarity between the corresponding data items.
    [Show full text]